PROFESSOR: We have the hydrogen atom Hamiltonian.
Hamiltonian.
And that was given by the kinetic operator
for the proton plus the kinetic operator
for the electron plus the potential, which
was a function of the distance between the proton
and the electron.
And what we achieved last time was the introduction
of two new pairs of canonical variables.
We had the electron position momentum, that's
a pair of canonical variables.
The proton position and momentum,
that's another pair of canonical variables.
They commute each pair, the two operators
commute to give IH bar, but the two pairs are independent.
So we search for another two pairs of variables,
and we found another two pairs.
One was the P and X associated with the center of mass motion,
and then we had the small p and small x associated
with the relative function.
And these four variables were a function of the original four
variables, the X and P of the electron
and the x and p of the proton.
So we define these two pairs, and they were canonical pairs.
This x with this p gave IH bar, this x with this p
gave IH bar, these p's and x's commute with any combination
of p's and x's over there.
But not only was that pretty good,
it simplified the Hamiltonian.
So at the end of the day, we had a Hamiltonian,
which was as if the center of mass moves like a free particle
plus a kinetic energy for the relative motion,
with a mass called the renews mass,
and the potential for the relative position.
And here, the mass, capital M, was the sum of the two masses.
And the relative mass was the product
of the masses over the sum, which
has the property that if one of the two masses
is much bigger than the other, it
gives you a mass mu proportional roughly equal to the lower
mass.
So this is the hydrogen atom reformulated.
And now, we want to write the Schrodinger equation
and just see effectively how the central potential formulation
of the relative motion arises.
Although, it starts to be a little somewhat clearer,
I think, that that's going to happen.
Another thing to notice of course,
is that if you're already thinking of a Schrodinger
equation, in which you will think
of the momenta as the derivative operators,
the center of mass momentum should
be thought as the derivative operator with respect
to the center of mass position.
So think of the center of mass as three coordinates,
and you differentiate with respect to them.
Similarly, for the relative momentum,
we'll think of it as a gradient with respect
to the relative position, because that's
the canonical coordinate that goes along with it.
That's how we should think of this operator as derivative.
And before, we didn't have to put those subscripts
in [INAUDIBLE],, because we always had just one coordinate
to work with.
But now, you have two coordinates.
What you've learned here in doing this analysis
was that we have a wave function that has coordinate dependence
on both the electron and the proton.
So let's do the separation of variables that shows
how to deal with this system.
So you want to write the wave function for the whole system,
so it depends on the center of mass
and on the relative coordinates.
Now, I don't put time, because we're
discussing time independent Schrodinger equation, where
the time you can put it later, if you
wish, with the total energy into the minus IET.
So what we will consider is a simple solution
that is of the product type.
So there will be a wave function associated up
factor, which is a wave function associated with center of mass,
and the wave function associated to the relative motion.
And we want to replace this into the Schrodinger equation
into H psi equals E psi.
So let's see how it would go.
Each term of the Hamiltonian is going
to act on this product of functions that determines
the whole wave function.
The center of mass momentum, being
derivative, respect to this x, will act just
on the first term.
So we'll have p squared over 2 m acting on psi c m of x.
And the relative wave function in that term
just goes for the ride.
So that's the first term in the Hamiltonian.
For the second term in the Hamiltonian,
we would have p squared over 2 mu.
We could add them psi relative, and let's put
even the potential here, v of x relative--
we don't put the relative on the x, but they'll just--
it's a small x psi relative of x.
So we've looked at the second term and the third term.
The third term is grouped with the second,
because it uses the little x, not the big X.
And then you have psi cm of x multiplicativly,
it doesn't do anything to it.
All this is equal to e times the psi cm
of x psi relative of little x.
OK, that's the Schrodinger equation.
And this should remind you, it's very similar to what
you did months ago of having motion's in say, in two
dimensions, and you wrote part of the wave function dependant
on x, part of a wave function dependant on y,
and you separated the Schrodinger equation.
So the next thing to do is to divide by the total wave
function, by the product.
So divide 5 by the total wave function.
So what do you get?
From the first term, you will get 1
over psi cm of capital X times this p squared over 2 m psi
cm of capital X. And that's all what comes of the first term.
From the second, you get plus 1 over psi relative
of x and this whole bracket squared
over 2 mu psi relative plus V of x psi relative.
And the whole thing being equal to E. The two wave functions
are divided there.
So we have a situation where a number
is the sum of two functions.
Now, what is funny of course, is the argument
you've heard several times.
This first term depends just on the capital X-coordinates.
The second term depends just on the lower x-coordinates,
small x-coordinates.
I didn't write them in some places, but here they are.
And therefore, the only way these two things can always
be true, is if the first term is a number.
And we'll call it Ecm.
The whole thing must be a number, we'll call it Ecm.
And the second term should be another number,
and I'll call it E relative.
That's E relative.
So our conclusion is that if this first term is
a number equal to Ecm we can multiply by psi cm
and get p squared over to 2m psi cm of x is
equal to Ecm times psi cm of x.
Which is a time independent Schrodinger equation
for a wave function psi cm that is moving freely.
The next equation is the one within brackets,
which is p squared over 2 mu psi relative of x plus v of--
now I can put r, when I say that r is the magnitude
of x psi relative of x equals e relative--
I should move this--
psi relative of x.
And we said that r was this.
So the second equation comes from this term
identified with relative.
And the last equation is to say that the total energy is
equal to Ecm plus e relative.
So the whole two body problem has been reduced
to these three equations.
This is what we aim to show.
This is a gradient squared on this wave function,
the central potential term, and the rest of the Schrodinger
equation.
So this is a Schrodinger equation
for a particle in the central potential.
That particle happens to be the relative distance
between these two particles, but it obeys a central equation
potential.
Here is the center of mass.
Solutions of this is a plane wave, momentum plane waves,
because this is like a free particle,
and that's our intuition.
The hydrogen atom can move like a free particle.
That's an overall quantum system and then
there's the relative degrees of freedom.
The total energy of this system must be the sum of the two.
So that said for the system, we are allowed now,
to consider the hydrogen atoms.
So that's what we'll do next.